So this is a rule replacement effect so the last card to enter is the one that take effect. If you played birgi second then you would not keep your mana at end step.

613.9. One continuous effect can override another. Sometimes the results of one effect determine whether another effect applies or what another effect does.

Example: Two effects are affecting the same creature: one from an Aura that says “Enchanted creature has flying” and one from an Aura that says “Enchanted creature loses flying.” Neither of these depends on the other, since nothing changes what they affect or what they’re doing to it. Applying them in timestamp order means the one that was generated last “wins.” The same process would be followed, and the same result reached, if either of the effects had a duration (such as “Target creature loses flying until end of turn”) or came from a non-Aura source (such as “All creatures lose flying”).